Conditions for tractability of the weighted $$L_p$$-discrepancy and integration in non-homogeneous tensor product spaces

Abstract We study tractability properties of the weighted $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -discrepancy, where the weights model the influence of different coordinates. A discrepancy is said to be tractable if the minimal number N of points, such that the discrepancy is less than the initial discrepancy times an error threshold $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> , does not grow exponentially fast with the dimension and $$\varepsilon ^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . In this case there are various notions of tractabilities used in order to classify the exact rate. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $$p \in (1, \infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and we prove a necessary condition for strong polynomial tractability. The proofs of the lower bounds are based on a powerful general result for the information complexity of integration with positive quadrature rules for tensor product spaces. As a second application of this general result we consider the integration of polynomials of degree at most 2.